Possibility of a hydrodynamic regime for electron transport was first
discussed by Gurzhi [1]}, [2]} for ultra-clean crystals
where electron-electron scattering, which conserves the momentum of
the electron fluid, dominates over the scattering of electrons
with impurities and with phonons which do not preserve
the momentum of electrons. As mentioned in the Introduction, this
remarkable possibility has been realized in
some ultra-clean systems (to suppress electron-impurity scattering)
at appropriately low temperatures (to suppress electron-phonon scatterings,
but still allowing significant electron-electron scattering). There have
been numerous investigations
[1]}, [2]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}
discussing constraints on impurity
concentration and the regime of temperature that make
electron-electron scattering dominant over the other, momentum
non-conserving, scattering modes. One also requires constraints on the system
size as scattering of electrons with the boundaries of the sample, in
general, leads to momentum loss from the electron fluid. With sufficient evidence
available for validity of this regime in these ultra-clean materials,
we assume validity of hydrodynamic description of electron
transport in these systems and write down the Navier-Stokes' equations
for the electron fluid. An important point we
mention here is that we restrict our discussion to the case of an inviscid
fluid. While experiments show that electron hydrodynamical flow has
non-zero viscosity, the viscous effects do not appear to be dominant compared
to other relevant effects like scattering of
electrons with impurities, phonons, and the sample boundary.
For certain cases, e.g. graphene, viscous effects may actually be
negligible [12]}.
